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<h3 class="heading"><span class="type">Paragraph</span></h3>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u_t = \alpha^2 u_{xx},\quad u(0,t) = a, ~~u(L,t) = b.
\end{equation*}
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<p class="continuation">We know that the steady state is <span class="process-math">\(U(x) = a + \frac{b-a}{L}x\text{.}\)</span> Now define a new variable</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w(x, t) = u(x, t) - U (x).
\end{equation*}
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<p class="continuation">Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w_t=u_t,\quad w_x=u_x-U'(x),\quad w(L,t)=u(L,t)-U(L)=b-b=0
\end{equation*}
</div>
<p class="continuation">which are homogeneous. Then, one can find the solution for <span class="process-math">\(w\)</span> by the standard separation of variables and Fourier series. Once this is done, one can go back to <span class="process-math">\(u\)</span> by</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=w(x,t)+U(x).
\end{equation*}
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<span class="incontext"><a href="sec7_8.html#p-423" class="internal">in-context</a></span>
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